Question

What is the remainder when ${1202}^{12}$ is divided by 1000?

• A. 120
• B. 1
• C. 0
• D. 296

Answer 1

The correct option is D

296

Write $(1202{)}^{12}=(1200+2{)}^{12}$. In this expansion there will be 13 terms.
From the binomial theorem, the $(r+1{)}^{th}$ term is given as $t_{r+1}{=}^n{C}_r{a}^{n-r}{b}^r$

The ${13}^{th}$ term is $2^{12}=4096$. ${12}^{th}$ term is ${}^{12}{C}_{11}\times {1200}^1\times 2^{11}=1\underset{–}{2}\times 1\underset{–}{2}00\times 204\underset{–}{8}=---200.$

When we add all ${13}^{th}$ and ${12}^{th}$ term, we get 296 as the last three digits.

The previous terms (1 to 11) will not make a difference to the answer as there will be a minimum of ${1200}^2$ involved in the other terms, therefore the terms will end with more than 3 zeroes.